Example 1. (Randomized response mechanism) 
Let \({\mathcal {X}}={\mathcal {Z}}=\lbrace 0,1\rbrace \)  and consider the mechanism given by the binary symmetric channel \(\mathsf {BSC}(\omega _\varepsilon )\)  with \(\omega _\varepsilon \frac{1}{1+e^\varepsilon }\) . This is often called randomized response mechanism [1]} and denoted by \({\mathsf {K}}^{\varepsilon }_{\mathsf {RR}}\) .
This simple mechanism is well-known to be \(\varepsilon \) -LDP which can now be verified via Theorem REF . Let \(P = {\mathsf {Bernoulli}}(p)\)  and \(Q={\mathsf {Bernoulli}}(q)\)  with \(p,q\in [0,1]\) . Then \(P{\mathsf {K}}^\varepsilon _{\mathsf {RR}} = {\mathsf {Bernoulli}}(p*\omega _\varepsilon )\)  and \(P{\mathsf {K}}^\varepsilon _{\mathsf {RR}} = {\mathsf {Bernoulli}}(q*\omega _\varepsilon )\)  where \(a*b a(1-b) + b(1-a)\) . It is straightforward to verify that \(|p*\omega _\varepsilon -e^\varepsilon q*\omega _\varepsilon |+|1-p*\omega _\varepsilon -e^\varepsilon (1-q*\omega _\varepsilon )| = 0.5(e^\varepsilon -1)\)  for any \(p,q\) , implying \({\mathsf {E}}_{e^\varepsilon }(P{\mathsf {K}}^\varepsilon _{\mathsf {RR}}\Vert Q{\mathsf {K}}^\varepsilon _{\mathsf {RR}}) = 0\) .
When \(|{\mathcal {X}}| = k\ge 2\) , a simple generalization of this mechanism, called \(k\) -ary randomized response, has been reported in literature (see, e.g., [2]}, [3]}) and is defined by \({\mathcal {Z}}= {\mathcal {X}}\)  and \({\mathsf {K}}_{\mathsf {kRR}}(x|x) = \frac{e^\varepsilon }{k-1+e^\varepsilon }\)  and \({\mathsf {K}}_{\mathsf {kRR}}(z|x) = \frac{1}{k-1+e^\varepsilon }\)  for \(z\ne x\) . Again, it can be verified that for this mechanism we have \({\mathsf {E}}_{e^\varepsilon }(P{\mathsf {K}}^\varepsilon _{\mathsf {kRR}}\Vert Q{\mathsf {K}}^\varepsilon _{\mathsf {kRR}}) = 0 \) , for all Bernoulli \(P\)  and \(Q\) .
